In how many different ways can the letters of the word 'OFFICE' be arranged so that the vowels never come together?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 288
Arrangements in which the vowels never come together are found by taking away, from all the arrangements, those in which the vowels do come together. OFFICE has 6 letters with F twice; its vowels are O, I and E.
All arrangements (F is repeated, so divide by \(2!\)):
\(\displaystyle\frac{6!}{2!}=\frac{720}{2}=360\)
Vowels together (O, I, E taken as one block, arranged with F, F and C):
The block and F, F, C: \(\displaystyle\frac{4!}{2!}=12\) ways
The vowels inside the block: \(3!=6\) ways
Together: \(12\times6=72\)
Vowels never together:
\(360-72=288\)
Tip: In such questions “the vowels never come together” means that the vowels are never all side by side; it does not mean that no two vowels touch.